What is a wave? The one-dimensional wave equation

2/3/14Mathematical description ofEM waves• Use of complex numbers to represent EMwaves• The complex refractive index– Scattering = real part– Absorption = imaginary part• Absorption and skin depth– Beer’s LawWhat is a wave?A wave is anything that moves.To displace any function f(x) to theright, just change its argument fromx to x-a, where a is a positivenumber.If we let a = v t, where v is positiveand t is time, then the displacementwill increase with time.So f(x - v t) represents a rightward,or forward, propagating wave.Similarly, f(x + v t) represents aleftward, or backward, propagatingwave, where v is the velocity of thewave.For an EM wave, we could have E = f(x ± vt)f(x) f(x-2)f(x-1) f(x-3)0 1 2 3x1

2/3/14The one-dimensional wave equationThe one-dimensional wave equation for scalar (i.e., non-vector)functions, f:where v will be the velocity of the wave.The wave equation has the simple solution:where f (u) can be any twice-differentiable function.What about a harmonic wave?E = E 0cos k(x − ct)E 0 = wave amplitude (related to the energycarried by the wave).€k = 2π = angular wavenumberλ = 2πν˜˜ ν(λ = wavelength; = wavenumber = 1/λ)€Alternatively:E = E 0cos(kx − ωt)€Where ω = kc = 2πc/λ = 2πf = angularfrequency (f = frequency)€2

2/3/14What about a harmonic wave?E = E 0cos k(x − ct);φ = k(x − ct)The argument of the cosine function represents the phase of the wave,ϕ, or the fraction of a complete cycle of the wave.€In-phase wavesLine of equal phase = wavefront = contours of maximum fieldOut-of-phasewavesThe Phase VelocityHow fast is the wave traveling?Velocity is a reference distancedivided by a reference time.The phase velocity is the wavelength / period: v = λ / τ Since f = 1/τ :v = λ fIn terms of k, k = 2π / λ, andthe angular frequency, ω = 2π / τ, this is:v = ω / k3

2/3/14The Group VelocityThis is the velocity at which the overall shape of the wave’s amplitudes,or the wave ‘envelope’, propagates. (= signal velocity)Here, phase velocity = group velocity (the medium is non-dispersive)Dispersion: phase/group velocity depends on frequencyBlack dot moves at phase velocity. Red dot moves at group velocity.This is normal dispersion (refractive index decreases with increasing λ)4

2/3/14Dispersion: phase/group velocity depends on frequencyBlack dot moves at group velocity. Red dot moves at phase velocity.This is anomalous dispersion (refractive index increases with increasing λ)Normal dispersion of visible lightShorter (blue) wavelengths refracted more than long (red) wavelengths.Refractive index of blue light > red light.5

2/3/14Complex numbersConsider a point,P = (x,y), on a 2DCartesian grid.Let the x-coordinate be the real partand the y-coordinate the imaginary partof a complex number.So, instead of using an ordered pair, (x,y), we write:P = x + i y= A cos(ϕ) + i A sin(ϕ)where i = √(-1)…or sometimes j = √(-1)Euler’s FormulaVoted the ‘Most Beautiful MathematicalFormula Ever’ in 1988Links the trigonometric functions and the complex exponential functionexp(iϕ) = cos(ϕ) + i sin(ϕ)so the point, P = A cos(ϕ) + i A sin(ϕ), can also be written:P = A exp(iϕ) = A e iφwhereA = Amplitudeϕ = Phase6

2/3/14Waves as rotating vectorsWaves using complex numbersE = E 0cos k(x − ct);φ = k(x − ct)The argument of the cosine function represents the phase of the wave,ϕ, or the fraction of a complete cycle of the wave.€Using complex numbers, we can write the harmonic wave equation as:E = E 0e ik(x −ct ) = E 0e i(kx−ωt )i.e., E = E 0 cos(ϕ) + i E 0 sin(ϕ), where the ‘real’ part of the expressionactually represents the wave.€We also need to specify the displacement E at x = 0 and t = 0, i.e., the‘initial’ displacement.7

2/3/14Amplitude and Absolute phaseE(x,t) = A cos[(k x – ω t ) – θ ]A = Amplitudeθ = Absolute phase (or initial, constant phase) at x = 0, t =0π kxWaves using complex numbersSo the electric field of an EM wave can be written:E(x,t) = E 0 cos(kx – ωt – θ)Since exp(iϕ) = cos(ϕ) + i sin(ϕ), E(x,t) can also be written:E(x,t) = Re { E 0 exp[i(kx – ωt – θ)] }Recall that the energy transferred by a wave (flux density) isproportional to the square of the amplitude, i.e., E 02 . Only theinteraction of the wave with matter can alter the energy of thepropagating wave.Remote sensing exploits this modulation of energy.8

2/3/14Waves using complex amplitudesWe can let the amplitude be complex:Where the constant stuff is separated from the rapidly changing stuff.€E(x,t) = E 0exp[i(kx − ωt − θ)]E(x,t) = [ E 0exp(−iθ) ]exp[i(kx − ωt)]The resulting "complex amplitude”:is constant in this case (as E 0 and θ are constant), which implies that themedium in which the wave is propagating is nonabsorbing.€[ E 0exp(−iθ) ]What happens to the wave amplitude upon interaction with matter?Vector fieldsHowever, light is a 3D vector field.A 3D vector field assigns a 3D vector (i.e., an arrow having bothdirection and length) to each point in 3D space.A light wave has both electric and magnetic 3D vector fields:And it can propagate in any direction, and point in any direction in space.9

2/3/14The 3D wave equation for the electric field and itssolutionA light wave can propagate in anydirection in space. So we must allowthe space derivative to be 3D:€€whose solution is: WhereAndE 0k = €E(x,y,z,t) = E 0exp(− k "⋅ x )exp([i( k '⋅ x − ωt)]is a constant, complex vectork '+ik "is a complex wave vector – the length of thisvector is inversely proportional to the wavelength of the wave.Its magnitude is the angular wavenumber, k = 2π/λ.x = (x,y,z)is a position vector€€€The 3D wave equation for the electric field and itssolution€E(x,y,z,t) = E 0exp(− k "⋅ x )exp([i( k '⋅ x − ωt)]The vector is normal to planes of constant phase (and henceindicates the direction of propagation of wave crests) k "The vector is normal to planes of constant amplitude. Note thatthese are not necessarily parallel. €k 'The amplitude of the wave at locationk "is now:So if is zero, then the medium is nonabsorbing, since theamplitude is constant . €€xE 0exp(−k "⋅x )10

2/3/14EM propagation in homogeneous materialsThe speed of an EM wave in free space is given by:ε 0 = permittivity of free space, µ 0 = magnetic permeability of free spaceTo describe EM propagation in other media,€two properties of the mediumare important, its electric permittivity ε and magnetic permeability µ.These are also complex parameters.ε = ε 0 (1+ χ) + i σ/ω = complex permittivityσ = electric conductivityχ = electric susceptibility (to polarization under the influence of anexternal field)Note that ε and µ also depend on frequency (ω).c =1µ 0ε 0= ω kEM propagation in homogeneous materialsIn a non-vacuum, the wave must still satisfy Maxwell’s Equations:v =1 µε = ω k€€We can now define the complex index of refraction, N, as the ratio ofthe wave velocity in free space to the velocity in the medium:N =µεµ 0ε 0= c v(n i= 0) or N = n r+ in iIf the imaginary part of N is zero, the material is nonabsorbing, and v isthe phase velocity of the wave in the medium. For most physical media,N > 1 (i.e., the speed of light is reduced relative to a vacuum).NB. N is a property of a particular medium and also a function of ω11

2/3/14EM propagation in homogeneous materialsRelationships between the wave vector and the refractive index (theseare derived from Maxwell’s Equations):k ʹ′ = ω n rc= 2π λReal part of wave vector€€k ʹ′ ʹ′ = ω n ic= 2πνn icImaginary part of wave vectorThese are the so-called ‘dispersion relations’ relating wavelength,frequency, velocity and refractive index.Absorption of EM radiationRecall the expression for the flux density of an EM wave (Poyntingvector):F = 1 2 cε 0 E 2When absorption occurs, the flux density of the absorbed frequencies isreduced.€12

2/3/14Absorption of EM radiationThe scalar amplitude of an EM wave at locationFrom the expression for the flux density we have:xF = F 0 [ exp(−k "⋅x )] 2 = F€ 0exp(−2k "⋅x )€is:E 0exp(−k "⋅x )€Absorption of EM radiationNow substitute the expression for :And we have:k ʹ′ ʹ′For a plane wave propagating in the x-direction.€F = F 0exp(− 4πνn ix)c€€k ʹ′ ʹ′ = ω n ic= 2πνn ic13

2/3/14Absorption coefficient and skin depthF = F 0exp(− 4πνn icx) = F 0e −β a xWhere β a is known as the absorption coefficient:The quantity 1/β a gives the distance required for the wave’s energy to be€attenuated to e -1 or ~37% of its original value, or the absorption/skindepth. It’s a function of frequency/wavelength.€4πνn i= 4πn ic λAbsorption coefficient and skin depthWithin a certain material, an EM wave with λ = 1 µm is attenuated to10% of its original intensity after propagating 1 cm. Determine theimaginary part of the refractive index n i .14

2/3/14The Forced Oscillator:The relative phaseof the oscillatormotion withrespect to theinput forcedepends on thefrequencies.Let the oscillator’s resonantfrequency be ω 0 , and theforcing frequency be ω.We could think of theforcing function as a lightelectric field and theoscillator as a nucleus ofan atom in a molecule.Belowresonanceω > ω 0ForceOscillatorWeakvibration.In phase.Strongvibration.90° out ofphase.Weakvibration.180° outof phase.Courtesy Prof. Rick Trebino, Georgia TechThe relative phaseof an electroncloud’s motionwith respect toinput light dependson the frequency.Belowresonanceω > ω 0Weakvibration.In phase.Courtesy Prof. Rick Trebino, Georgia Tech15

2/3/14The relativephase ofemitted lightwith respect tothe input lightdepends on thefrequency.The emitted light is90° phase-shiftedwith respect to theatom’s motion.Belowresonanceω > ω 0Electric fieldat atomElectroncloudEmittedfieldWeakemission.90° out ofphase.Strongemission.180° outof phase.Weakemission.-90° outof phase.Courtesy Prof. Rick Trebino, Georgia TechRefractive index (n) – the dispersion equationn = 1+q 2e2ε 0m∑kq e = charge on an electronε 0 = electric constantm = mass of an electronN k € = number of charges (oscillators) of type k per unit volumeω = angular frequency of the EM radiationω k = resonant frequency of an electron bound in an atomγ = ‘damping coefficient’ for oscillator k (oscillation cannot be permanent)What is the refractive index of visible light in air?What happens as the frequency of EM radiation increases at constant ω k ?What happens if the resonant frequency is in the visible range?What happens if ω > ω k ? e.g., shine x-rays on glass, or radio waves on freeelectrons.N kω k2− ω 2 + iγ kω16

2/3/14Refractive index (n) of water and icePenetration depth of water and ice(also called absorption depth or skin depth)17

2/3/14Refractive index and colorPaluweh volcanoAqua MODIS: Flores, Indonesia; Feb 2, 2013 (250 meters)Color of suspended sedimentAqua MODIS: Flores, Indonesia; Jan 23, 2013 (250 meters)18

2/3/14Complex refractive index of volcanic ashFrom: Krotkov et al. (1997), Ultraviolet optical model of volcanic clouds for remotesensing of ash and sulfur dioxide. J. Geophys. Res., 102 (D18), 21891-21904.19